Aclass has 32 students. in how many different ways can five students form a group for an activity? (assume the order of the students is not important.)

This is a combination problem (35C5)
In general, it is the number of arrangements of 5 out of 35 (35P5) divided by the number of arrangements of 5 out of 5 (5P5), which reduces to 
35P5/5P5 = 35!/((30-5)!)  /  (5!/(5-5)!) = 35!/(5!(30-5)!) = 35C5.

Substituting values
35C5 = 35!/(5!30!)=35*34*33*32*31/(1*2*3*4*5) = 324632 ways

The correct answer is: b

a. geometricb. g[h] = (3/4)·g[h-1]; g[1] = 384c. g[h] = 384·(3/4)^(h-1)d. g[12] ≈ 16.2183 gallonse. g[12] ≈ 64.8732 quarts

step-by-step explanation:

part a. the differences are decreasing, so we know the sequence is not arithmetic. the ratios are constant (=3/4), so we know the sequence is geometric.

part b. from part a, we know the common ratio is 3/4, so each term is 3/4 of the one before. the initial term is the number of gallons at h=1.

g[h] = (3/4)·g[h-1]

g[1] = 384 . . . . initial value required for a recursive definition

part c. the general formula for a geometric sequence is

a[n] = a[1]·r^(n-1)

we have a[1] = 384, r = 3/4, and the term number given by h. the formula that follows is

g[h] = 384·(3/4)^(h-1)

part d. filling in the value and doing the arithmetic, we have

g[12] = 384·(3/4)^11 ≈ 384·0.042235136 ≈ 16.2183 . . gallons

part e. each gallon is 4 quarts, so

16.2183 gal × 4 qt/gal = 64.8732 qt